A Proposal for Community (Quasi-) Monte Carlo Software Fred J. Hickernell Department of Applied Mathematics Center for Interdisciplinary Scientific Computation Illinois Institute of Technology [email protected] mypages.iit.edu/~hickernell Thanks to the workshop organizers, the GAIL team NSF-DMS-1522687 and NSF-DMS-1638521 (SAMSI) Follow-Up to Discussions, June 28, 2018
Motivation Landscape Big Decisions Next Steps Elements References Questions that I Ask or Hear Asked Where can I find quality, free quasi-Monte Carlo (qMC) software? 2/28
Motivation Landscape Big Decisions Next Steps Elements References Questions that I Ask or Hear Asked Where can I find quality, free quasi-Monte Carlo (qMC) software? Where can I find use cases that illustrate how to employ qMC methods? 2/28
Motivation Landscape Big Decisions Next Steps Elements References Questions that I Ask or Hear Asked Where can I find quality, free quasi-Monte Carlo (qMC) software? Where can I find use cases that illustrate how to employ qMC methods? How can I try that qMC method developed by X’s group for my problem? 2/28
Motivation Landscape Big Decisions Next Steps Elements References Questions that I Ask or Hear Asked Where can I find quality, free quasi-Monte Carlo (qMC) software? Where can I find use cases that illustrate how to employ qMC methods? How can I try that qMC method developed by X’s group for my problem? How can I try my qMC method on the example shown by Y’s group? 2/28
Motivation Landscape Big Decisions Next Steps Elements References Questions that I Ask or Hear Asked Where can I find quality, free quasi-Monte Carlo (qMC) software? Where can I find use cases that illustrate how to employ qMC methods? How can I try that qMC method developed by X’s group for my problem? How can I try my qMC method on the example shown by Y’s group? How can my student get results without writing code from scratch? 2/28
Motivation Landscape Big Decisions Next Steps Elements References Questions that I Ask or Hear Asked Where can I find quality, free quasi-Monte Carlo (qMC) software? Where can I find use cases that illustrate how to employ qMC methods? How can I try that qMC method developed by X’s group for my problem? How can I try my qMC method on the example shown by Y’s group? How can my student get results without writing code from scratch? How can the code that I use benefit from recent developments? 2/28
Motivation Landscape Big Decisions Next Steps Elements References Questions that I Ask or Hear Asked Where can I find quality, free quasi-Monte Carlo (qMC) software? Where can I find use cases that illustrate how to employ qMC methods? How can I try that qMC method developed by X’s group for my problem? How can I try my qMC method on the example shown by Y’s group? How can my student get results without writing code from scratch? How can the code that I use benefit from recent developments? How can my work receive wider recognition? 2/28
Motivation Landscape Big Decisions Next Steps Elements References Questions that I Ask or Hear Asked Where can I find quality, free quasi-Monte Carlo (qMC) software? Where can I find use cases that illustrate how to employ qMC methods? How can I try that qMC method developed by X’s group for my problem? How can I try my qMC method on the example shown by Y’s group? How can my student get results without writing code from scratch? How can the code that I use benefit from recent developments? How can my work receive wider recognition? My initial attempts in this direction over the past 5 years have produced GAIL Choi, S.-C. T. et al. GAIL: Guaranteed Automatic Integration Library (Versions 1.0–2.2). MATLAB software. 2013–2017. http://gailgithub.github.io/GAIL_Dev/. 2/28
Motivation Landscape Big Decisions Next Steps Elements References Can We Have qMC Community Software that Grows Up to Be Like ... Chebfun Computing with Chebyhsev polynomials, chebfun.org Clawpack Solution of conservation laws, clawpack.org deal.II Finite-elements, http://dealii.org Mission: To provide well-documented tools to build finite element codes for a broad variety of PDEs, from laptops to supercomputers. Vision: To create an open, inclusive, participatory community providing users and developers with a state-of-the-art, comprehensive software library that constitutes the go-to solution for all finite element problems. FEniCS Finite-elements, fenicsproject.org Gromacs Molecular dynamics, gromacs.org Stan Markov Chain Monte Carlo, mc-stan.org Trilinos Multiphysics computations, trilinos.org Developed and supported by multiple research groups Used beyond the research groups that develop it A recognized standard in its field 3/28
Motivation Landscape Big Decisions Next Steps Elements References What Is Available Now John Burkhardt Variety of qMC Software in C++, Fortran, MATLAB, and Python, people.sc.fsu.edu/~jburkardt/ Mike Giles Multi-Level Monte Carlo Software in C++, MATLAB, Python, and R, people.maths.ox.ac.uk/gilesm/mlmc/ Fred Hickernell Guaranteed Automatic Integration Library (GAIL) in MATLAB, gailgithub.github.io/GAIL_Dev/ Stephen Joe & Frances Kuo Sobol’ generators in C++, Generating vectors for lattices, web.maths.unsw.edu.au/~fkuo/ Pierre L’Ecuyer Random number generators, Stochastic Simulation, Lattice Builder in C/C++ and Java, simul.iro.umontreal.ca Dirk Nuyens Magic Point Shop, QMC4PDE, etc. in MATLAB, Python, and C++, people.cs.kuleuven.be/~dirk.nuyens/ Art Owen Various code, statweb.stanford.edu/~owen/code/ Christoph Schwab Partial differential equations with random coefficients MATLAB Sobol’ and Halton sequences Python Sobol’ and Halton sequences R randtoolbox Sobol’, lattice, and Halton sequences 4/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Key Elements Sequences—IID, Sobol’, lattice, Halton, sparse grid, ..., including randomization; fixed and extensible sample size and dimension; constructions using optimization 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Key Elements Sequences—IID, Sobol’, lattice, Halton, sparse grid, ..., including randomization; fixed and extensible sample size and dimension; constructions using optimization Sequence generators—with inputs d, coordinate indices, and index range 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Key Elements Sequences—IID, Sobol’, lattice, Halton, sparse grid, ..., including randomization; fixed and extensible sample size and dimension; constructions using optimization Sequence generators—with inputs d, coordinate indices, and index range Discrepancy measures—various kernels and domains 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Key Elements Sequences—IID, Sobol’, lattice, Halton, sparse grid, ..., including randomization; fixed and extensible sample size and dimension; constructions using optimization Sequence generators—with inputs d, coordinate indices, and index range Discrepancy measures—various kernels and domains Variable transformations to accommodate non-uniform distributions and domains other than the unit cube 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Key Elements Sequences—IID, Sobol’, lattice, Halton, sparse grid, ..., including randomization; fixed and extensible sample size and dimension; constructions using optimization Sequence generators—with inputs d, coordinate indices, and index range Discrepancy measures—various kernels and domains Variable transformations to accommodate non-uniform distributions and domains other than the unit cube Integrands, but some will come from external library, e.g., PDE solvers 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Key Elements Sequences—IID, Sobol’, lattice, Halton, sparse grid, ..., including randomization; fixed and extensible sample size and dimension; constructions using optimization Sequence generators—with inputs d, coordinate indices, and index range Discrepancy measures—various kernels and domains Variable transformations to accommodate non-uniform distributions and domains other than the unit cube Integrands, but some will come from external library, e.g., PDE solvers Integrators, including multilevel and multivariate decomposition methods 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Key Elements Sequences—IID, Sobol’, lattice, Halton, sparse grid, ..., including randomization; fixed and extensible sample size and dimension; constructions using optimization Sequence generators—with inputs d, coordinate indices, and index range Discrepancy measures—various kernels and domains Variable transformations to accommodate non-uniform distributions and domains other than the unit cube Integrands, but some will come from external library, e.g., PDE solvers Integrators, including multilevel and multivariate decomposition methods Stopping criteria 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Key Elements Sequences—IID, Sobol’, lattice, Halton, sparse grid, ..., including randomization; fixed and extensible sample size and dimension; constructions using optimization Sequence generators—with inputs d, coordinate indices, and index range Discrepancy measures—various kernels and domains Variable transformations to accommodate non-uniform distributions and domains other than the unit cube Integrands, but some will come from external library, e.g., PDE solvers Integrators, including multilevel and multivariate decomposition methods Stopping criteria Compelling use cases 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Key Elements Sequences—IID, Sobol’, lattice, Halton, sparse grid, ..., including randomization; fixed and extensible sample size and dimension; constructions using optimization Sequence generators—with inputs d, coordinate indices, and index range Discrepancy measures—various kernels and domains Variable transformations to accommodate non-uniform distributions and domains other than the unit cube Integrands, but some will come from external library, e.g., PDE solvers Integrators, including multilevel and multivariate decomposition methods Stopping criteria Compelling use cases Packages that display output in tables or plots 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Languages and Architectures Which one(s)? Python? C++? 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Languages and Architectures Which one(s)? Python? C++? How do we balance performance, developer time, and portability? 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Languages and Architectures Which one(s)? Python? C++? How do we balance performance, developer time, and portability? How will users connect the software with other software packages and environments? 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Languages and Architectures Which one(s)? Python? C++? How do we balance performance, developer time, and portability? How will users connect the software with other software packages and environments? How will parallel computing be supported? 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Good Development Practices Start small, with good skeleton 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Good Development Practices Start small, with good skeleton Version control on Git or equivalent 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Good Development Practices Start small, with good skeleton Version control on Git or equivalent Ownership of routines, updates require owners’ approval 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Good Development Practices Start small, with good skeleton Version control on Git or equivalent Ownership of routines, updates require owners’ approval Comprehensive tests run regularly 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Good Development Practices Start small, with good skeleton Version control on Git or equivalent Ownership of routines, updates require owners’ approval Comprehensive tests run regularly Reasonable license 5/28
Motivation Landscape Big Decisions Next Steps Elements References Decisions to Make If We Want to Succeed Good Development Practices Start small, with good skeleton Version control on Git or equivalent Ownership of routines, updates require owners’ approval Comprehensive tests run regularly Reasonable license Marketing on websites and at conferences 5/28
Motivation Landscape Big Decisions Next Steps Elements References Bigger than One Research Group We have experience over the past 5 years developing GAIL Our group has learned some of the discipline required to develop good software Our group does not have the capacity to tackle this whole project, and neither does your group A good software library should attract developers Let’s leave a legacy to our community that goes beyond theorems and algorithms Choi, S.-C. T. et al. GAIL: Guaranteed Automatic Integration Library (Versions 1.0–2.2). MATLAB software. 2013–2017. http://gailgithub.github.io/GAIL_Dev/. 6/28
Motivation Landscape Big Decisions Next Steps Elements References We Must Weigh ... Costs Benefits Less time to prove theorems More impact for our theorems Learning a new language Wider access and better performance for our code Compromise with other research groups More capable code than can be produced by one research group Time spent writing documentation and tests Fewer bugs for those who use the code Attract more qMC developers Attract more qMC users Happier funding agencies 7/28
Motivation Landscape Big Decisions Next Steps Elements References Current Discussions If you are interested, whether you are a potential developer or user, let’s talk 8/28
Motivation Landscape Big Decisions Next Steps Elements References Current Discussions If you are interested, whether you are a potential developer or user, let’s talk Determine what we can agree on as initial answers to the big questions 8/28
Motivation Landscape Big Decisions Next Steps Elements References Current Discussions If you are interested, whether you are a potential developer or user, let’s talk Determine what we can agree on as initial answers to the big questions Right now we are trying to focus on 8/28
Motivation Landscape Big Decisions Next Steps Elements References Current Discussions If you are interested, whether you are a potential developer or user, let’s talk Determine what we can agree on as initial answers to the big questions Right now we are trying to focus on An initial language 8/28
Motivation Landscape Big Decisions Next Steps Elements References Current Discussions If you are interested, whether you are a potential developer or user, let’s talk Determine what we can agree on as initial answers to the big questions Right now we are trying to focus on An initial language An initial version control platform, and a model for collaboration 8/28
Motivation Landscape Big Decisions Next Steps Elements References Current Discussions If you are interested, whether you are a potential developer or user, let’s talk Determine what we can agree on as initial answers to the big questions Right now we are trying to focus on An initial language An initial version control platform, and a model for collaboration A few initial routines 8/28
Motivation Landscape Big Decisions Next Steps Elements References Current Discussions If you are interested, whether you are a potential developer or user, let’s talk Determine what we can agree on as initial answers to the big questions Right now we are trying to focus on An initial language An initial version control platform, and a model for collaboration A few initial routines An initial use case 8/28
Motivation Landscape Big Decisions Next Steps Elements References Current Discussions If you are interested, whether you are a potential developer or user, let’s talk Determine what we can agree on as initial answers to the big questions Right now we are trying to focus on An initial language An initial version control platform, and a model for collaboration A few initial routines An initial use case We welcome you to join us. Email me to join our Google group 8/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Discrete Distributions classdef (Abstract) discreteDistribution %Specifies and generates the components of an n ÿ i=1 wi δxi (¨) properties (Abstract) distribData %information required to generate the distribution state %state of the generator nStreams end properties domain = [0 0; 1 1]; %domain of the discrete distribution, X domainType = 'box' %domain of the discrete distribution, X dimension = 2 %dimension of the domain, d trueDistribution = 'uniform' %name of the distribution that the discrete ... distribution attempts to emulate end 9/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Discrete Distributions classdef (Abstract) discreteDistribution methods (Abstract) genDistrib(obj, nStart, nEnd, n, coordIndex) % nStart = starting value of i % nEnd = ending value of i % n = value of n used to determine an % coordIndex = which coordinates in sequence are needed end end 10/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Discrete IID Uniform classdef IIDDistribution < discreteDistribution %Specifies and generates the components of 1 n n ÿ i=1 δxi (¨) %where the xi are IID uniform on [0, 1]d or IID standard Gaussian properties distribData %stream data state = [] %not used nStreams = 1 end methods function obj = initStreams(obj,nStreams) obj.nStreams = nStreams; obj.distribData.stream = ... RandStream.create('mrg32k3a','NumStreams',nStreams,'CellOutput',true); end 11/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Discrete IID Uniform classdef IIDDistribution < discreteDistribution function [x, w, a] = genDistrib(obj, nStart, nEnd, n, coordIndex, ... streamIndex) if nargin < 6 streamIndex = 1; end nPts = nEnd ´ nStart + 1; %how many points to be generated if strcmp(obj.trueDistribution, 'uniform') %generate uniform points x = ... rand(obj.distribData.stream{streamIndex},nPts,numel(coordIndex)); ... %nodes else %standard normal points x = ... randn(obj.distribData.stream{streamIndex},nPts,numel(coordIndex)); ... %nodes end w = 1; a = 1/n; end end 12/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Functions classdef (Abstract) fun % Specify and generate values f(x) for x P X properties domain = [0 0; 1 1] %domain of the function, X domainType = 'box' %e.g., 'box', 'ball' dimension = 2 %dimension of the domain, d distribType = 'uniform' %e.g., 'uniform', 'Gaussian' nominalValue = 0 %a nominal number, c, such that (c, . . . , c) P X end methods (Abstract) y = f(obj, xu, coordIndex) % xu = nodes, xu,i = ith row of an n ˆ |u| matrix % coordIndex = set of those coordinates in sequence needed, u % y = n ˆ p matrix with values f(xu,i , c) where if x1 i = (xi,u , c)j, then x1 ij = xij for j P u, and x1 ij = c otherwise end end 13/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Keister’s Function classdef KeisterFun < fun % Specify and generate values f(x) for x P X methods function y = f(obj, x, coordIndex) %if the nominalValue = 0, this is efficient normx2 = sum(x.*x,2); if (numel(coordIndex) ‰ obj.dimension) && (obj.nominalValue ‰ 0) normx2 = normx2 + (obj.nominalValue.^2) * (obj.dimension ´ ... numel(coordIndex)); end y = exp(´normx2) .* cos(sqrt(normx2)); end end end 14/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—StoppingCriteria classdef (Abstract) stoppingCriterion % Decide when to stop a properties absTol = 1e´2 %absolute tolerance, d relTol = 0 %absolute tolerance, d nInit = 1024 %initial sample size nMax = 1e8 %maximum number of samples allowed end properties (Abstract) discDistAllowed %which discrete distributions are supported decompTypeAllowed %which decomposition types are supported end methods (Abstract) stopYet(obj, distribObj) % distribObj = data or summary of data computed already end end 15/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—CLTStopping classdef CLTStopping < stoppingCriterion % Stopping criterion based on the Central Limit Theorem properties discDistAllowed = "IIDDistribution" %which discrete distributions are ... supported decompTypeAllowed = ["single"; "multi"] %which decomposition types are ... supported inflate = 1.2 %inflation factor alpha = 0.01; end properties (Dependent) quantile end 16/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—CLTStopping classdef CLTStopping < stoppingCriterion methods function [obj, dataObj, distribObj] = ... stopYet(obj, dataObj, funObj, distribObj) if numel(dataObj) dataObj = meanVarData; end switch dataObj.stage 17/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—CLTStopping classdef CLTStopping < stoppingCriterion switch dataObj.stage case 'begin' %initialize dataObj.timeStart = tic; if any(strcmp(obj.discDistAllowed,class(distribObj))) error('Stoppoing criterion not compatible with sampling ... distribution') end nf = numel(funObj); %number of functions whose integrals add up ... to the solution distribObj = initStreams(distribObj,nf); %need an IID stream ... for each function dataObj.prevN = zeros(1,nf); %initialize data object dataObj.nextN = repmat(obj.nInit,1,nf); dataObj.muhat = Inf(1,nf); dataObj.sighat = Inf(1,nf); dataObj.nSigma = obj.nInit; %use initial samples to estimate ... standard deviation dataObj.costF = zeros(1,nf); dataObj.stage = 'sigma'; %compute standard deviation next 18/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—CLTStopping classdef CLTStopping < stoppingCriterion case 'sigma' dataObj.prevN = dataObj.nextN; %update place in the sequence tempA = sqrt(dataObj.costF); %use cost of function values to ... decide how to allocate tempB = sum(tempA .* dataObj.sighat); %samples for computation ... of the mean nM = ceil((tempB*(obj.quantile*obj.inflate ... /max(obj.absTol,dataObj.solution*obj.relTol))^2) ... * (dataObj.sighat./sqrt(dataObj.costF))); dataObj.nMu = min(max(dataObj.nextN,nM),obj.nMax ´ dataObj.prevN); dataObj.nextN = dataObj.nMu + dataObj.prevN; dataObj.stage = 'mu'; %compute sample mean next 19/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—CLTStopping classdef CLTStopping < stoppingCriterion case 'mu' dataObj.solution = sum(dataObj.muhat); dataObj.nSamplesUsed = dataObj.nextN; errBar = (obj.quantile * obj.inflate) * ... sqrt(sum(dataObj.sighat.^2/dataObj.nMu)); dataObj.errorBound = dataObj.solution + errBar*[´1 1]; dataObj.stage = 'done'; %finished with computation end dataObj.timeUsed = toc(dataObj.timeStart); end function value = get.quantile(obj) value = ´norminv(obj.alpha/2); end end end 20/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Integration function [solution, dataObj] = integrate(funObj, distribObj, stopCritObj) %Specify and generate values f(x) for x P X % funObj = an object from class fun % distribObj = an object from class discrete_distribution % stopcritObj = an object from class stopping_criterion %Initialize the accumData object and other crucial objects [stopCritObj, dataObj, distribObj] = stopYet(stopCritObj, [], funObj, ... distribObj); while strcmp(dataObj.stage, 'done') %the dataObj.stage property tells us ... where we are in the process dataObj = updateData(dataObj, distribObj, funObj); %compute additional data [stopCritObj, dataObj] = stopYet(stopCritObj, dataObj, funObj); %update ... the status of the computation end solution = dataObj.solution; %assign outputs dataObj.timeUsed = toc(dataObj.timeStart); 21/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Integration Example %% How to integrate a function using our community QMC framework % An example with Keister's function integrated with respect to the uniform % distribution over the unit cube stopObj = CLTStopping %stopping criterion for IID sampling using the ... Central Limit Theorem distribObj = IIDDistribution; %IID sampling with uniform distribution [sol, out] = integrate(KeisterFun, distribObj, stopObj) >> IntegrationExample sol = 0.4310 out = timeUsed: 0.0014 nSamplesUsed: 7540 errorBound: [0.4210 0.4410] 22/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Integration Example stopObj.absTol = 1e´3 %decrease tolerance [sol, out] = integrate(KeisterFun, distribObj, stopObj) sol = 0.4253 out = timeUsed: 0.0333 nSamplesUsed: 652546 errorBound: [0.4243 0.4263] 23/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Integration Example stopObj.absTol = 0; %impossible tolerance stopObj.nMax = 1e6; %calculation limited by sample budget [sol, out] = integrate(KeisterFun, distribObj, stopObj) sol = 0.4252 out = timeUsed: 0.0392 nSamplesUsed: 1000000 errorBound: [0.4244 0.4260] 24/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Asian Call, Low D %A multilevel example of Asian option pricing distribObj.trueDistribution = 'normal'; %Change to normal distribution stopObj.absTol = 0.01; %increase tolerance stopObj.nMax = 1e8; %pushing the sample budget back up OptionObj = AsianCallFun(4) %4 time steps [sol, out] = integrate(OptionObj, distribObj, stopObj) OptionObj = dimension: 4 sol = 6.1740 out = timeUsed: 1.0517 nSamplesUsed: 5680546 errorBound: [6.1640 6.1840] 25/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Asian Call, High D OptionObj = AsianCallFun(64) %single level, 64 time steps [sol, out] = integrate(OptionObj, distribObj, stopObj) OptionObj = AsianCallFun with properties: dimension: 64 sol =6.2036 out = meanVarData with properties: timeUsed: 25.1910 nSamplesUsed: 5610402 errorBound: [6.1936 6.2136] 26/28
Motivation Landscape Big Decisions Next Steps Elements References Elements of the Community Software—Asian Call, Multi-Level OptionObj = AsianCallFun([4 4 4]) %multilevel, 64 time steps, faster [sol, out] = integrate(OptionObj, distribObj, stopObj) OptionObj = 1×3 AsianCallFun array with properties: sol = 6.2052 out = timeUsed: 2.2171 nSamplesUsed: [8080862 446720 85907] errorBound: [6.1968 6.2135] 27/28
Motivation Landscape Big Decisions Next Steps Elements References Choi, S.-C. T. et al. GAIL: Guaranteed Automatic Integration Library (Versions 1.0–2.2). MATLAB software. 2013–2017. http://gailgithub.github.io/GAIL_Dev/. 28/28
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